1,735 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
On rainbow tetrahedra in Cayley graphs
Let be the complete undirected Cayley graph of the odd cyclic
group . Connected graphs whose vertices are rainbow tetrahedra in
are studied, with any two such vertices adjacent if and only if they
share (as tetrahedra) precisely two distinct triangles. This yields graphs
of largest degree 6, asymptotic diameter and almost all vertices
with degree: {\bf(a)} 6 in ; {\bf(b)} 4 in exactly six connected subgraphs
of the -semi-regular tessellation; and {\bf(c)} 3 in exactly four
connected subgraphs of the -regular hexagonal tessellation. These
vertices have as closed neighborhoods the union (in a fixed way) of closed
neighborhoods in the ten respective resulting tessellations. Generalizing
asymptotic results are discussed as well.Comment: 21 pages, 7 figure
Tverberg plus constraints
Many of the strengthenings and extensions of the topological Tverberg theorem
can be derived with surprising ease directly from the original theorem: For
this we introduce a proof technique that combines a concept of "Tverberg
unavoidable subcomplexes" with the observation that Tverberg points that
equalize the distance from such a subcomplex can be obtained from maps to an
extended target space.
Thus we obtain simple proofs for many variants of the topological Tverberg
theorem, such as the colored Tverberg theorem of Zivaljevic and Vrecica (1992).
We also get a new strengthened version of the generalized van Kampen-Flores
theorem by Sarkaria (1991) and Volovikov (1996), an affine version of their
"j-wise disjoint" Tverberg theorem, and a topological version of Soberon's
(2013) result on Tverberg points with equal barycentric coordinates.Comment: 15 pages; revised version, accepted for publication in Bulletin
London Math. Societ
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